Central limit theorem

Irrespective of theshape of the distribution of the population or universe, the distribution of average values of samples drawn from that universe will tend toward a normal distribution as the sample size grows without bound.

It can also be shown that the average of sampleaverages will equal the average of the universe and that the standard deviation of the averages equals the standard deviation of the universe divided by the square root of the sample size. Shewhart performed experiments that showed that small sample sizeswere needed to get approximately normal distributions from even wildly non-normal universes. Figure IV.3 was created by Shewhart using samples of four measurements.

Used by permission of the publisher, ASQC Quality Press. Milwaukee, Wisconsin.

The practical implications of the central limit theorem are immense. Consider that without the central limit theorem effects, we would have to develop a separate statistical model for every non-normal distribution encountered in practice. This would be the only way to determine if the system were exhibiting chance variation. Because of the central limit theorem we can use averages of small samples to evaluate any process using the normal distribution. The central limit theorem is the basis for the most powerful of statistical process control tools, Shewhart control charts.

Since 1982: The art & science to improve your bottom line

Quality America
offers Statistical Process Control software, as well as training materials for Lean Six
Sigma, Quality Management and SPC. We embrace a customer-driven approach, and lead in
many software innovations, continually seeking ways to provide our customers with the
best and most affordable solutions. Leaders in their field, Quality America has provided
software and training products and services to tens of thousands of companies in over
25 countries.